"We shall not cease from exploration
And the end of all our exploring
Will be to arrive where we started
And know the place for the first time."
—from Little Gidding by T.S. Eliot
If you took calculus in high school or college, you might remember the chain rule. One of the main topics in calculus is learning how to approximate functions with simpler functions. The chain rule tells you how to do this for functions that depend on other functions. It helps you keep track of the way a small change in a variable will change the value of one function, which in turn changes the value of another function, and so on in a long chain of functions. It's an important method, but it's one of the more challenging topics in a calculus class. It's even more challenging when instead of one variable, you're trying to keep track of several variables that affect several outputs, as we do in my multivariable calculus class.
When I started preparing for teaching the chain rule in my class, I didn't like the way the book did it. I felt like the proof was overly complicated and seemed to deliberately avoid an easier method. I couldn't figure out why the author approached the proof the way he did. I started writing up different notes that I thought would be more clear. After a few hours of work, I realized that what made my method easier was that I assumed a simple case of the chain rule. Theorems are easier to prove when you assume they're true in the first place!
I started to fix my mistake and discovered that I was also using slightly more restrictive assumptions, so my proof works for some functions but not for all functions. The method the author of the book chose works for functions that my method couldn't tackle. I finally understood why he had done it that way in the first place!
Frustrated with myself, this phrase from Little Gidding popped into my head: "the end of all our exploring will be to arrive where we started and know the place for the first time." T.S. Eliot wasn't writing about math, but this sentiment seemed perfect for my situation, and for the way I feel about math in general a lot of the time. Only after extensive exploration do I fully understand the beauty or utility of earlier theorems or methods!
When I talked with mathematicians Laura DeMarco and Amie Wilkinson last spring, both of them mentioned how much more they appreciate calculus now than they did when they first learned it. Calculus is far from a basic topic in math, but it is one that is learned very early, before we understand much about just how intricate it is. We do a lot of exploring before we come back and truly understand calculus for the first time. Of course, mathematicians are not unique in feeling this way. I'm only an amateur musician, but the more music I play and listen to, the more I feel like I can understand Bach for the first time. I'm sure the same thing happens in other subjects.
I did not end up using the book's presentation of the chain rule in my class. But now I understand why the author did. The method I decided to use is also more involved than I thought it would be when I started working on it. It is one of the more complicated proofs we are doing in this class, and I know some of my students are frustrated. I hope that they see our class as an exploration. And as we keep exploring, moving beyond the chain rule to some of its consequences and applications, I hope they will be able to come back and feel like they know the chain rule for the first time.